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Introduction to Neutrosophic Statistics

Florentin Smarandache

arXiv:1406.2000v1cs.AI

TL;DR

Neutrosophic statistics had remained largely undeveloped despite its earlier definition and publication. This paper develops the concept through multiple approaches, examples, and generalizations, including neutrosophic extensions of statistical measures and distributions.

  • Problem

    Neutrosophic statistics had been defined and published but remained largely undeveloped, motivating further development of statistics for indeterminate data.

  • Method

    The paper presents several neutrosophic-statistics approaches through examples and generalizes them to classes of examples, including ordered observations and distributions.

  • Results

    The paper develops neutrosophic measures and distributions, including a binomial distribution extension and a multinomial generalization for trials with indeterminacy.

  • Takeaways & Limitations

    Neutrosophic statistics can be developed through multiple approaches because different problems involve different types of indeterminacy.

  • Takeaways & Limitations

    The distribution examples assume a fixed number of independent trials with the same outcome probabilities for each outcome.

Abstract

from arXiv · show

Neutrosophic Statistics means statistical analysis of population or sample that has indeterminate (imprecise, ambiguous, vague, incomplete, unknown) data. For example, the population or sample size might not be exactly determinate because of some individuals that partially belong to the population or sample, and partially they do not belong, or individuals whose appurtenance is completely unknown. Also, there are population or sample individuals whose data could be indeterminate. In this book, we develop the 1995 notion of neutrosophic statistics. We present various practical examples. It is possible to define the neutrosophic statistics in many ways, because there are various types of indeterminacies, depending on the problem to solve.

Introduction · Neutrosophic Statistics

Neutrosophic Statistics extends classical statistics by analyzing data containing indeterminacy through set-valued representations and specialized methods. The sections define its scope, motivate the transition from crisp values, and present flexible approaches for organizing, interpreting, and visualizing such data.

  • Introduction: Classical equations can be adapted by replacing indeterminate numbers with sets and applying set operations, with aN denoting an imprecise or indeterminate value.When no indeterminacy exists, aN = a.
  • Introduction: The framework addresses quantities and sample membership that cannot be computed or specified exactly in real-life statistical problems.For example, a sample size may lie between 90 and 100 when some individuals only partially belong to the population or their membership is unknown.
  • Introduction: Neutrosophic Statistics extends classical statistics by replacing determined crisp data with values or sets containing indeterminacy.Indeterminacy may be ambiguous, vague, imprecise, incomplete, or unknown.
  • Neutrosophic Statistics: Neutrosophic Statistics comprises data and analytical methods designed to organize and interpret indeterminate data to reveal underlying patterns.The approach can address both randomness and indeterminacy, unlike classical statistics, which addresses determined data and randomness.
  • Neutrosophic Statistics: The framework supports multiple approaches, illustrated through examples and generalized to classes of examples rather than restricted to one method.The text explicitly leaves room for readers to invent additional approaches.
  • Neutrosophic Statistics: A neutrosophical statistical number separates a determinate part d from an indeterminate part i, as N = d + i.For example, a = 5 + i with i ∈ [0, 0.4] is equivalent to a ∈ [5, 5.4].
  • Neutrosophic Statistics: Neutrosophic frequency distributions represent categories and frequencies with indeterminate limits, producing intervals for relative frequencies and totals.The example accumulates relative-frequency intervals to [0.859, 1.152].
  • Neutrosophic Statistics: Neutrosophic graphs can represent indeterminacy in 2D or add a third dimension whose axis measures the data’s indeterminate component.The 3D example uses coordinates whose second component is determinate and third component is maximum indeterminacy.

Neutrosophic Quartiles · Neutrosophic Sample · Neutrosophic Numerical

The sections define neutrosophic quartiles for partially ordered observation sets and characterize neutrosophic samples through indeterminate membership, data, or sample composition. They also illustrate how standard sampling designs become neutrosophic when indeterminacy is present.

  • Neutrosophic Quartiles: Neutrosophic observations are arranged in almost ascending order because sets have only a partial order.A total order can be defined by comparing midpoints, then minimums, with maximums coinciding when both prior values are equal.
  • Neutrosophic Quartiles: Neutrosophic quartiles follow classical rank definitions, using the 1/4(n + 1)th and 2/4(n + 1)th positions and analogous upper-quartile placement.When a rank falls between observations, the method averages the two neutrosophic observations; another procedure uses the inferior integer part.
  • Neutrosophic Sample: A neutrosophic sample is a population subset containing indeterminacy among individuals or in the subset as a whole, yielding vague or incomplete information.A neutrosophic population likewise has uncertain membership, and an element may be t % in the population, f % outside it, and i % indeterminate.
  • Neutrosophic Sample: A simple random neutrosophic sample of size n contains at least one individual with some indeterminacy.In the laptop example, 100 homes have single nonworking laptops, producing a simple random neutrosophic sample of size 100.
  • Neutrosophic Sample: Stratified and cluster sampling become neutrosophic when sampled individuals or cluster members have indeterminate classifications or research-topic affiliations.Examples include transgender individuals discovered within gender strata and graduate students undecided between classical and neutrosophic statistics.
  • Neutrosophic Sample: Neutrosophic statistics also addresses uncertainty about which sample respondents provide unreliable or malicious data and therefore should be removed.This creates indeterminacy in the sample size and in how the affected data should be represented or excluded.

Measures · Classical Neutrosophic · Numbers

The merged sections define classical neutrosophic numbers, develop their arithmetic and division conditions, and illustrate neutrosophic dispersion through a worked standard-deviation example. They show how indeterminacy is represented algebraically and propagated through calculations.

  • Measures: For the four numbers −2 −4𝐼, −1 + 0 ∙𝐼, 3 + 5𝐼, and 6 + 7𝐼, deviations are computed from the neutrosophic mean 1.5 + 2𝐼.The reported deviations are −3.5 −6𝐼, −2.5 −2𝐼, 1.5 + 3𝐼, and 4.5 + 5𝐼.
  • Measures: Squaring the deviations uses 𝐼2 = 𝐼, so indeterminacy propagates through the variance calculation.The worked terms include 12.25 + 42𝐼 + 36𝐼 and 2.25 + 18𝐼.
  • Measures: 3.20 + 0.64𝐼 is the neutrosophic standard deviation of the four numbers, with 3.20 equal to the classical standard deviation of their determinate parts.The passage states that 0.64 is not the classical standard deviation of the indeterminate parts −4, 0, 5, and 7.
  • Numbers: A classical neutrosophic number has the form a + b𝐼, with real or complex coefficients, where 𝐼 represents indeterminacy and satisfies 0 ∙𝐼 = 0 and 𝐼2 = 𝐼.Also, 𝐼n = 𝐼 for every positive integer n.
  • Numbers: With real coefficients, a + b𝐼 is a Neutrosophic Real Number; with complex coefficients, it is a Neutrosophic Complex Number.A complex form can be written as a + bi + c𝐼 + di𝐼, with real a, b, c, and d.
  • Numbers: A real number may be represented as a degenerated neutrosophic number, whereas a true neutrosophic number has a non-zero coefficient for 𝐼.For example, 5 = 5 + 0 ∙𝐼 is degenerated.
  • Division of classical neutrosophic real numbers: Division of classical neutrosophic real numbers has a unique solution only when a2(a2 + b2) ≠ 0, equivalently a2 ≠ 0 and a2 ≠ −b2.These conditions arise from requiring the determinant of the coefficient system to be nonzero.
  • Division of classical neutrosophic real numbers: Division by I, −I, or k𝐼 is undefined, while some other divisors yield infinitely many solutions rather than a unique quotient.For the infinite-solution case, x ∈ ℛ and y = 1 − x.

Root index 𝒏≥𝟐 of a neutrosophic real number.

The section computes roots of neutrosophic real numbers by expressing a root in neutrosophic form, raising it to the relevant power, and solving the resulting algebraic system. For roots of index n, the method distinguishes neutrosophic real from complex solutions according to whether the auxiliary variables are real or complex.

  • Square roots: For square roots, writing √(a + bI) = x + yI and squaring produces x^2 = a and 2xy + y^2 = b.The equations follow from I^2 = I.
  • Square roots: The example √(9 + 7I) yields four solutions: ±3 ± I.The corresponding ordered pairs are (3, −7), (3, 1), (−3, 7), and (−3, −1).
  • Roots of index n: For roots of index n, expanding (x + yI)^n and setting the real component to zero gives x = 0 and n solutions for the roots of 1.These comprise the real solution y = 1 and n−1 complex solutions.
  • Roots of index n: The same procedure applies to any neutrosophic number of root index n ≥ 2 by solving the resulting equations for x and y.Real x and y produce neutrosophic real solutions, whereas complex x and y produce neutrosophic complex solutions.
  • Neutrosophic complex roots: For a neutrosophic complex number, raising x + yi + zI + wiI to the nth power yields a nonlinear algebraic system in four variables and four equations.The equations equate the real, i, I, and iI components to the corresponding coefficients.

Neutrosophic Random · Numbers

Neutrosophic Random Numbers can be generated from a pool of sets rather than only crisp numbers. Random extraction and replacement then produce a sequence of intervals, with crisp numbers as a special case.

  • Numbers: Neutrosophic Random Numbers may be generated using a pool of sets instead of only crisp numbers.The method uses 100 balls, each labeled with an interval [a, b].
  • Numbers: The pool can contain 100 balls, each labeled with an interval [a, b].The endpoints satisfy a, b ∈ {1, 2, 3, …, 100} and a ≤ b.
  • Numbers: Each interval endpoint is selected from {1, 2, 3, …, 100}.The construction also requires a ≤ b.
  • Numbers: When a = b, the interval [a, a] represents the crisp number a.Thus, crisp numbers are included within the interval-based pool.
  • Numbers: When a < b, the labeled object represents the set [a, b].The construction therefore distinguishes singleton intervals from broader sets.
  • Numbers: A ball is randomly extracted, its interval is registered, and the ball is returned to the pool.This extraction-and-replacement process is repeated.
  • Numbers: Repeated sampling produces a random sequence of intervals rather than a random sequence of crisp numbers.The distinction follows from labeling the balls with intervals or sets.

Example with Neutrosophic Data · the sample size · Neutrosophic Binomial

The examples represent uncertain observations as intervals and derive interval-valued descriptive statistics. When a sample may contain an unknown erroneous observation, all possible reduced samples can be combined using interval, average, or weighted-average procedures.

  • Example with Neutrosophic Data: Two observations are indeterminate, represented as intervals [2, 5] and [18, 24] rather than exact values.The four observations are rewritten uniformly as [6, 6], [2, 5], [30, 30], and [18, 24].
  • Example with Neutrosophic Data: The median lies in the interval [16, 17.5], while the average lies in [14, 16.25].These results are obtained by applying interval arithmetic to the ordered observations.
  • Example with Neutrosophic Data: Squared deviations are also computed as intervals, including [64, 104.04], [190.44, 256], and other interval-valued terms.The calculations extend interval arithmetic from central-tendency measures to dispersion-related quantities.
  • the sample size: When one of five observations is certainly wrong but unidentified, the median belongs to [8.5, 10.5].The observations are reordered as 5, 8, 9, 12, and 17 before studying all possibilities.
  • the sample size: The mean belongs to [8.5, 11.5], and the standard deviation belongs to [2.5, 4.43706].These intervals combine the results from the possible samples formed after accounting for the unknown erroneous observation.
  • the sample size: Weighted averaging assigns each possible sample a weight representing its chance of being the correct sample.The example uses weights 0.4, 0.1, 0.3, 0.2, and 0.7; the combined metrics are therefore inclined toward the fifth sample.
  • the sample size: For n observations with k wrong observations, a program studies the C(n,n-k) samples formed by discarding k observations and calculates each sample’s statistical metrics.The resulting C(n,n-k) results can then be combined using interval, average, weighted-average, or other procedures.

Distribution · Neutrosophic Multinomial Distribution

The paper extends the binomial distribution to trials with success, failure, and indeterminacy, with indeterminacy handled through a problem-dependent threshold. It then generalizes this framework to r possible outcomes plus indeterminacy, yielding a neutrosophic multinomial distribution.

  • Distribution: Neutrosophic binomial distribution extends the classical binomial model by allowing indeterminacy in the probabilistic experiment.Each trial can produce success, failure, or indeterminacy.
  • Distribution: The model assumes a fixed number of independent trials with identical chances of success, failure, and indeterminacy.The neutrosophic binomial random variable counts successes across n≥1 trials.
  • Distribution: Indeterminacy across n trials is defined using a threshold for the number of indeterminate outcomes, whose interpretation depends on the problem and expert viewpoint.Outcomes above the threshold are treated as indeterminate, while outcomes at or below it are determinate.
  • Distribution: The distribution represents exactly x successes among n trials with a three-component neutrosophic probability vector: determinate success, indeterminacy, and alternative determinate outcomes.The framework permits complete, incomplete, and paraconsistent probabilities depending on the sum of outcome chances.
  • Distribution: In the watch example, the assigned chances are P(F)=0.8, P(S)=0.1, and P(I)=0.2, producing paraconsistent probability because their sum is 1.1.The example considers analog-display watches among the next five purchases with indeterminacy threshold 2.
  • Neutrosophic Multinomial Distribution: The neutrosophic multinomial distribution generalizes the binomial case to r≥2 possible outcomes at each trial, together with an indeterminacy outcome.Its expansion is (P1 + P2 + ⋯ + Pr + i)^n for n trials.
  • Neutrosophic Multinomial Distribution: The multinomial framework assigns probabilities to counts α1 through αr and β indeterminate events satisfying α1 + α2 + ⋯ + αr + β = n.These counts arise from n independent trials, and Xj records how often event Ej occurs.

Neutrosophic Scatter Plot … Squares Lines

Neutrosophic graphical and regression methods represent indeterminate data with points, line segments, surfaces, or higher-dimensional objects rather than only classical numerical forms. Neutrosophic least-squares lines retain classical formulas while replacing numbers with sets, producing interval-valued coefficients, predictions, and residuals.

  • Neutrosophic Scatter Plot: A neutrosophic scatter plot contains at least one not-well-defined point, with coordinates represented by intervals or sets rather than precise numbers.For example, (3, 5) is precise, whereas coordinates such as ([2, 4), 7) and (, [5, 7]) are imprecise.
  • Neutrosophic Scatter Plot: Bivariate neutrosophic scatter plots may include points, line segments, surfaces, or parts of these geometrical objects.In general, n-variate plots formed from n − 1 independent variables and one dependent variable can contain objects of dimensions 0 through n.
  • Neutrosophic Least-: A neutrosophic function has at least one indeterminate coefficient or independent-variable value, and its graph generally has higher dimension than the corresponding classical graph.For example, a classical curve can become a surface, while a classical surface can become a bigger surface or solid.
  • Neutrosophic Regression: Neutrosophic regression analyzes associations between neutrosophic independent and dependent variables to formulate equations or formulas for predicting future dependent values.The regression graph may be a thick or strip curve because neutrosophic theory handles indeterminacy and approximations.
  • Neutrosophic Regression: Neutrosophic regression may be linear or nonlinear, including parabolic, elliptic, and hyperbolic second-degree forms.The classification follows whether the association between independent and dependent variables is linear or nonlinear.
  • Squares Lines: Neutrosophic least-squares lines use the classical formula but replace numerical data and coefficients with sets, so the intercept and slope may be sets.The intercept is a = ȳ − b x̄, using neutrosophic averages, and predicted values are denoted with a circumflex accent.
  • Squares Lines: The example neutrosophic least-squares line is ŷ = (−22.2157, 5.61905) + (0.42857, 6.58824)x, representing a geometrical surface between two lines.The coefficients are interval-valued, so plotted predictions are segments rather than single points.
  • Squares Lines: Each real value in the example belongs to or is included in its predicted value interval, illustrating interval-valued neutrosophic predictions and residuals.The reported inclusions include y2 = 6 ∈ (−20.5014, 38.5603) and y4 = (10, 13) ⊂ (−19.6643, 51.7367).

Neutrosophic Coefficient · A Neutrosophic Normal Distribution

The section defines neutrosophic measures for linear association and extends normal distributions by allowing imprecise parameters. It illustrates interval-based uncertainty in distribution ranges and generalizes the approach to many classical distributions.

  • of Determination: The Neutrosophic Coefficient of Determination represents the proportion of variation in y explained by an approximate linear relationship between x and y.
  • of Determination: 60.37%–62.74% of sample variation is explained by the neutrosophic approximate linear relationship between x and y.
  • of Determination: The neutrosophic correlation coefficient extends Pearson’s coefficient to neutrosophic data by using sets rather than individual numbers.Its formula remains the classical one, with neutrosophic covariance and sample standard deviations.
  • of Determination: Neutrosophic weighted random numbers generalize random numbers by assigning each number x_j a different chance p_j to occur.
  • A Neutrosophic Normal Distribution: A neutrosophic normal distribution is a classical normal distribution whose mean, standard deviation, variance, or combination is imprecise, often represented by intervals.
  • A Neutrosophic Normal Distribution: 68% of values lie in x∈[12, 18] when μ = 15 and σ = [2, 3], while 97,7% lie in x∈[6, 24].The area between the lowest and highest curves represents graph burden, or indeterminacy.
  • A Neutrosophic Normal Distribution: With μ = [15, 17] and σ = [2, 3], double indeterminacy produces ranges of [12, 20], [9, 23], and [6, 26] for approximately 68%, 95.4%, and 97.7% of values.
  • 𝑥∈[6, 26]. Neutrosophication of Other Distributions.: Replacing one or more classical distribution parameters with sets extends standard normal, bivariate normal, uniform, sampling, geometric, hypergeometric, Poisson, chi-squared, exponential, Pareto, and t-distributions to neutrosophic versions.A replacing set may contain two or more elements or be empty when the parameter is unknown.

A Neutrosophic Hypothesis

A Neutrosophic Hypothesis concerns neutrosophic values of one or more population characteristics, including indeterminate, unclear, vague, unknown, or inexact values. Testing NH0 against NHa retains the classical reject-or-fail-to-reject structure while accommodating neutrosophic errors, probabilities, and test statistics.

  • A Neutrosophic Hypothesis: A Neutrosophic Hypothesis states claims about the neutrosophic values of one or more population characteristics.The characteristics may include indeterminate values, several unknown values, or an inexact number of terms for discrete variables.
  • A Neutrosophic Hypothesis: NH0 is initially assumed true, whereas NHa is the alternative hypothesis.Testing compares NH0 with NHa and leads to rejecting NH0 when evidence strongly suggests it is false, or failing to reject NH0 when evidence is insufficient.
  • Errors: Neutrosophic Type I Error rejects NH0 when it is true, while Neutrosophic Type II Error fails to reject NH0 when it is false.Inference from a neutrosophic sample characteristic to a population characteristic is subject to error because a census may be difficult or impossible.
  • Errors: αN and βN denote the probabilities of neutrosophic Type I and Type II errors, respectively, and may be subsets of [0, 1].The ideal procedure has αN = βN ≡ 0 or tiny intervals near zero; αN = [0.07, 0.10] corresponds to rejecting a true NH0 about 7, 8, 9, or 10 times in a hundred, while βN = [0.07, 0.10] corresponds to accepting a false NH0 about 7-10 times in a hundred.
  • Testing: In a neutrosophic mean test, n, x̅, and s can be sets, and for n > 30 the statistic z has an approximately neutrosophic standard normal distribution.For the exam-anxiety example, n = 64, x̅ = [48.0, 50.0], and s = 25; with α = 0.10, z = [2.24, 3.20] exceeds the one-tailed critical value 1.28, so NH0 is rejected and the mean score is higher than 41.0.

The Neutrosophic Level of Significance · The Neutrosophic Confidence Interval

Neutrosophic significance levels, P-values, and confidence intervals represent uncertain quantities as sets or intervals rather than crisp numbers. The sections define corresponding decision rules, interval constructions, and illustrative numerical results.

  • The Neutrosophic Level of Significance: A neutrosophic significance level α may be a set, such as α4 = [0.01, 0.10], rather than a crisp number.The interval indicates that α varies across [0.01, 0.10].
  • The Neutrosophic Level of Significance: A neutrosophic P-value is the smallest significance level at which H0 can be rejected, but it is a set or interval rather than a crisp number.It is computed as the classical probability of observing a more extreme test statistic assuming H0 is true.
  • The Neutrosophic Level of Significance: Decision-making compares neutrosophic P-values with α or αN: non-overlapping ranges support rejection or non-rejection, while overlap indicates indeterminacy.Intersecting P-value and significance-level sets also permits calculating the chances of rejecting and not rejecting H0.
  • The Neutrosophic Confidence Interval: A neutrosophic confidence interval is an interval of plausible neutrosophic characteristic values containing the characteristic with a chosen confidence level.Its confidence level describes the percentage of successful samples whose intervals include the population characteristic.
  • The Neutrosophic Confidence Interval: Classical confidence-interval formulas extend to neutrosophic variables, using σ when known and s when σ is unknown with sample size exceeding 30.Critical values 1.645, 1.96, and 2,58 correspond to 90%, 95%, and 99% confidence, respectively.
  • The Neutrosophic Confidence Interval: In the vision-loss example, combining the interval cases produces the neutrosophic confidence interval [16.94, 21.06].The construction begins with a sample of 60 people whose average loss is 18%-20% and standard deviation is 4%-5%.

Large-Sample … The Neutrosophic Central Limit Theorem

The section extends classical large-sample inference to neutrosophic settings where sample sizes, proportions, and critical values may be sets. It establishes the neutrosophic central limit theorem and illustrates both proportion and small-sample t confidence intervals.

  • Interval for the Population Proportion: Neutrosophic proportion intervals allow p, n, and the z critical value to be sets rather than crisp numbers.The z critical value may be [1.645, 1.96], corresponding to confidence levels of [90, 95]%.
  • Interval for the Population Proportion: A survey of 200–220 consumers with 150 yes responses gives the point estimate p approximately [0.68, 0.75].The set-valued sample size represents uncertainty about whether 20 people belonged to the dealer’s customer population.
  • Interval for the Population Proportion: The resulting conservative large-sample neutrosophic confidence interval for the population proportion is [0.590626, 0.839374].The calculation uses z critical value 2.58 and combines interval endpoints conservatively.
  • The Neutrosophic Central Limit Theorem: The neutrosophic central limit theorem applies safely when min{n} exceeds 30 and approximates the sampling distribution of x̅ by a neutrosophic normal curve regardless of population shape.For a normal population, min{n} may be below 30 and the sampling distribution remains normal for any neutrosophic sample size.
  • The Neutrosophic Central Limit Theorem: This theorem enables large-sample neutrosophic procedures for inferring a population mean even when the population distribution’s shape is unknown.When the population is nonnormal, larger min{n} yields a better normal approximation.
  • The Neutrosophic Central Limit Theorem: The theorem does not apply when min{n} is small and the population distribution is unknown; a neutrosophic t interval instead requires a normal or approximately normal population.The neutrosophic t interval accommodates set-valued x̅, s, and n.
  • The Neutrosophic Central Limit Theorem: For 18 workers, the 95% neutrosophic t confidence interval for the population mean weight lifted is [6.011, 11.989] kg.The interval uses 17 degrees of freedom and combines the neutrosophic sample mean [8, 10] kg with standard deviation [3, 4] kg.
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